Math 54 - Quiz 07 Ans, Spring 2006

Math 54 - Quiz 07 Ans, Spring 2006 - -8 µ-3 [4-( λ-1)] =...

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Math 54, Quiz 7 Solutions Sections 202/203 GSI Carter 1. Let A and B be any n × n matrices with an eigenvalue of 2. For each of the following statements, write the word “true” or “false.” (a) The matrix A 2 must have an eigenvalue of 4. True. See exercise 23 in section 5.2. (b) The matrix AB must have an eigenvalue of 4. False. Consider A = ± 2 0 0 1 ² and B = ± 1 0 0 2 ² . 2. Find any nonzero eigenvector for the following matrix and its associated eigenvalue: 1 0 3 2 1 4 - 1 2 1 Let A denote the given matrix. Then the characteristic polynomial of A is det( λI - A ) = ³ ³ ³ ³ ³ ³ λ - 1 0 - 3 - 2 λ - 1 - 4 1 - 2 λ - 1 ³ ³ ³ ³ ³ ³ = ( λ - 1) ´ ( λ - 1) 2
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Unformatted text preview: -8 µ-3 [4-( λ-1)] = ( λ-1)( λ 2-2 λ-7) + 3( λ-5) = λ 3-2 λ 2-7 λ-λ 2 + 2 λ + 7 + 3 λ-15 = λ 3-3 λ 2-2 λ-8 = ( λ-4)( λ 2 + λ + 2) . Therefore λ = 4 is an eigenvalue. To find an eigenvector, we need to compute the null space of 4 I-A = 3-3-2 3-4 1-2 3 . This row reduces to 1-2 3-1 2 6-12 and then to 1-1 1-2 . Therefore the vector (1 , 2 , 1) is an eigenvector for A with eigenvalue 4. 1...
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